Triangle calculator SSA

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Triangle has two solutions with side c=9.6443904215 and with side c=1.84767624318

#1 Acute scalene triangle.

Sides: a = 7.5   b = 6.2   c = 9.6443904215

Area: T = 23.2466183019
Perimeter: p = 23.3443904215
Semiperimeter: s = 11.67219521075

Angle ∠ A = α = 51.0388226456° = 51°2'18″ = 0.8910785096 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 88.9621773544° = 88°57'42″ = 1.55326758568 rad

Height: ha = 6.19989821384
Height: hb = 7.49987687158
Height: hc = 4.82109070726

Median: ma = 7.18774852524
Median: mb = 8.06333395224
Median: mc = 4.90985413183

Inradius: r = 1.99216276905
Circumradius: R = 4.82327438633

Vertex coordinates: A[9.6443904215; 0] B[0; 0] C[5.74553333234; 4.82109070726]
Centroid: CG[5.13297458461; 1.60769690242]
Coordinates of the circumscribed circle: U[4.82219521075; 0.08773856033]
Coordinates of the inscribed circle: I[5.47219521075; 1.99216276905]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.9621773544° = 128°57'42″ = 0.8910785096 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 91.0388226456° = 91°2'18″ = 1.55326758568 rad




How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 7.5 ; ; b = 6.2 ; ; c = 9.64 ; ;

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 7.5+6.2+9.64 = 23.34 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.34 }{ 2 } = 11.67 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.67 * (11.67-7.5)(11.67-6.2)(11.67-9.64) } ; ; T = sqrt{ 540.39 } = 23.25 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 23.25 }{ 7.5 } = 6.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 23.25 }{ 6.2 } = 7.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 23.25 }{ 9.64 } = 4.82 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 7.5**2-6.2**2-9.64**2 }{ 2 * 6.2 * 9.64 } ) = 51° 2'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.2**2-7.5**2-9.64**2 }{ 2 * 7.5 * 9.64 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.64**2-7.5**2-6.2**2 }{ 2 * 6.2 * 7.5 } ) = 88° 57'42" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 23.25 }{ 11.67 } = 1.99 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.5 }{ 2 * sin 51° 2'18" } = 4.82 ; ;





#2 Obtuse scalene triangle.

Sides: a = 7.5   b = 6.2   c = 1.84767624318

Area: T = 4.45215350344
Perimeter: p = 15.54767624318
Semiperimeter: s = 7.77333812159

Angle ∠ A = α = 128.9621773544° = 128°57'42″ = 2.25108075576 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 11.0388226456° = 11°2'18″ = 0.19326533952 rad

Height: ha = 1.18770760092
Height: hb = 1.43659790434
Height: hc = 4.82109070726

Median: ma = 2.62196880997
Median: mb = 4.49766949796
Median: mc = 6.81985311564

Inradius: r = 0.57326639297
Circumradius: R = 4.82327438633

Vertex coordinates: A[1.84767624318; 0] B[0; 0] C[5.74553333234; 4.82109070726]
Centroid: CG[2.53106985851; 1.60769690242]
Coordinates of the circumscribed circle: U[0.92333812159; 4.73435214694]
Coordinates of the inscribed circle: I[1.57333812159; 0.57326639297]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 51.0388226456° = 51°2'18″ = 2.25108075576 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 168.9621773544° = 168°57'42″ = 0.19326533952 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 7.5 ; ; b = 6.2 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 6.2**2 = 7.5**2 + c**2 -2 * 6.2 * c * cos (40° ) ; ; ; ; c**2 -11.491c +17.81 =0 ; ; p=1; q=-11.4906666468; r=17.81 ; ; D = q**2 - 4pr = 11.491**2 - 4 * 1 * 17.81 = 60.7954199875 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 11.49 ± sqrt{ 60.8 } }{ 2 } ; ; c_{1,2} = 5.74533332339 ± 3.89857089161 ; ; c_{1} = 9.643904215 ; ;
c_{2} = 1.84676243179 ; ; ; ; (c -9.643904215) (c -1.84676243179) = 0 ; ; ; ; c>0 ; ;


Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 7.5 ; ; b = 6.2 ; ; c = 1.85 ; ;

2. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 7.5+6.2+1.85 = 15.55 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15.55 }{ 2 } = 7.77 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.77 * (7.77-7.5)(7.77-6.2)(7.77-1.85) } ; ; T = sqrt{ 19.82 } = 4.45 ; ;

5. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.45 }{ 7.5 } = 1.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.45 }{ 6.2 } = 1.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.45 }{ 1.85 } = 4.82 ; ;

6. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 7.5**2-6.2**2-1.85**2 }{ 2 * 6.2 * 1.85 } ) = 128° 57'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.2**2-7.5**2-1.85**2 }{ 2 * 7.5 * 1.85 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.85**2-7.5**2-6.2**2 }{ 2 * 6.2 * 7.5 } ) = 11° 2'18" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.45 }{ 7.77 } = 0.57 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.5 }{ 2 * sin 128° 57'42" } = 4.82 ; ;




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